Spaces of commuting elements in the classical groups

Abstract

Let G be the classical group, and let Hom(Z m , G) denote the space of commuting m-tuples in G. First, we refine the formula for the Poincaré series of Hom(Z m , G) due to Ramras and Stafa by assigning (signed) integer partitions to (signed) permutations. Using the refined formula, we determine the top term of the Poincaré series, and apply it to prove the dependence of the topology of Hom(Z m , G) on the parity of m and the rational hyperbolicity of Hom(Z m , G) for m ≥ 2. Next, we give a minimal generating set of the cohomology of Hom(Z m , G) and determine the cohomology in low dimensions. We apply these results to prove homological stability for Hom(Z m , G) with the best possible stable range. Baird proved that the cohomology of Hom(Z m , G) is identified with a certain ring of invariants of the Weyl group of G, and our approach is a direct calculation of this ring of invariants.

Mixed Hodge structures on character varieties of nilpotent groups

“…Remark 9. The characteristic exponents referenced in Theorem 8 coincide with the ones for a maximal compact K ⊂ G. Therefore, these are well-known for all simple G (see [RS,Table 1] or the table in [KT,Page 7]).…”

Torsion in the space of commuting elements in a Lie group

“…and so it is often called the space of commuting m-tuples in G. The topology of Hom(Z m , G) has been intensely studied in recent years. See for example [1,2,3,4,5,11,13,17,20,21]. In this paper, we study the homology of Hom(Z m , G).…”

“…In [4], Baird described the cohomology of Hom(Z m , G) 1 over a field F in terms of a certain ring of invariants the Weyl group of G. Based on this description, Ramras and Stafa [20] (cf. [17]) gave a formula for the Poincaré series of the cohomology of Hom(Z m , G) 1 over a field F. Then the rank of the homology of Hom(Z m , G) 1 was determined by setting F = Q. So we next study: Problem 1.1.…”